If you memorize some of the "nice" log10 approximations like

  ...
  log10 2   ~= 0.3
  log10 2.5 ~= 0.4
  log10 3   ~= 0.5
  log10 4   ~= 0.6
  ...

then 25000^(1/8) could be done mentally as

  log10 of 25000 is 4.4
  4.4 / 8 is 0.55
  antilog10 of 0.55 is approximately 3.5